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As in A072868 described by OEIS; Defined by $\sigma(\sigma(n)-n) = n$.

Since these numbers are important in regard to many things, specially mersenne primes, since ${n-1 \over 2}\times n,~\sigma(\sigma(n)-n) = n$.

Expression explained: $n$ minus $1$ divided by $2$ is a perfect number whenever $\sigma(\sigma(n)-n) = n$

Is there any known way to calculate $n$ for $\sigma(\sigma(n)-n) = n$, or is this yet to be discovered?

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  • $\begingroup$ The OEIS link already says that it is conjectured the only such n are Mersenne primes. Is your question whether we know how to generate Mersenne primes? You will notice by reading the FAQ that the questions usually welcome here are the ones for which you hope to get some help from experts, not well known open problems. So please clarify if you're looking for some heuristics or something similar. $\endgroup$
    – Gjergji Zaimi
    Jan 23, 2013 at 1:50
  • $\begingroup$ Hi! Thanks for your answer! I am afraid that you are getting me wrong, as my question is not directly in regard to the primes, but whether there is an existing way of getting $n=f(x)$ where $\sigma(\sigma(n)-n) = n$ $\endgroup$
    – JohnWO
    Jan 23, 2013 at 2:21
  • $\begingroup$ @Gjergji, nitpick: it is conjectured that the only such $n$ are $n=p+1$ with $p$ a Mersenne prime. But, JohnWO, that means the only known way to get such $n$ is to find Mersenne primes and then subtract $1$. If anyone finds an $n$ by some other method, it will either mean the conjecture is false or that we now have a new method for finding Mersenne primes. $\endgroup$
    – Gerry Myerson
    Jan 23, 2013 at 4:32
  • $\begingroup$ Gerry Myerson: Thanks for the answer! So basically you're saying: If $n = f(x)$ where $n= \sigma(\sigma(n)-n)$ with x NOT being a Mersenne prime, it would disprove the conjecture? Please correct me if wrong. $\endgroup$
    – JohnWO
    Jan 23, 2013 at 6:07
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    $\begingroup$ I don't know what you mean by $f(x)$. It is conjectured at OEIS that if $\sigma(\sigma(n)-n)=n$ then $n-1$ is a Mersenne prime. So, if $\sigma(\sigma(n)-n)=n$ and $n-1$ is not a Mersenne prime then, yes, that would disprove the conjecture at OEIS. $\endgroup$
    – Gerry Myerson
    Jan 23, 2013 at 11:56

1 Answer 1

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Here, as my contribution, I add these conjectures (I don't know if these are in the literature) for a subset of the lattice $\mathbb{Z}_{\geq 1}\times\mathbb{Z}_{\geq 1}$. My computational evidence is small, I hope that it is interesting for you or for the professors and users of this site. Of course I don't know how solve the question.

As usual we denote the sum of positive divisors function of an integer $n\geq 1$ as $\sigma(n)=\sum_{1\leq d\mid n}d$, that is an important multiplicative function. I add the Wikipedia Divisor function.

Conjecture 1. If $x\geq 2$ and $y\geq 1$ are both integers, and satisfy $$\sigma(\sigma(xy)-xy)=x,\tag{1}$$ then $y=1$.

Remark. In previous conjecture, and it applies also for the following conjectures, one can to express the consequence of the conjecture (I say from the computational evidence that I got) as $x-1$ is a Mersenne prime instead of the claimed consequence/conjecture $y=1$.

Conjecture 2. If $x\geq 2$ and $y\geq 1$ are both integers, and these numbers satisfy $$\sigma(\sigma(xy)-(x+y-1))=x,\tag{2}$$ then $y=1$.

Conjecture 3. Let $x\geq 2$ and $y\geq 1$ be integers such that $\sigma(x+y-1)>xy$. If the equation $$\sigma(\sigma(x+y-1)-xy)=x,\tag{3}$$ holds, then $y=1$.

I refer below the reference for the equation, and I add that one knows (if it can be inspiring here) also the article Variations on Euclid's formula for perfect numbers, by Farideh Firoozbakht and Maximilian F. Hasler, from Journal of Integer Sequences (2010) Volume: 13, Issue: 3, Article 10.3.1.

References:

[1] Sequence A072868, from The On-Line Encyclopedia of Integer Sequences.

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  • $\begingroup$ Please if some professor/user finds a counterexample it is welcome that he/she comment it, many thanks. I add for example the script written in Pari/GP for(x=2, 1000, for(y=1, 1000, if(sigma(sigma(x*y)-(x+y-1))==x,print(x-1," ",y)))) that you can to evaluate from the web Sage Cell Server, just choose GP as language. I add as reference the PARI/GP Developers group of Université Bordeaux 1. In a similar way you can to evaluate after let's say a minute this script for(x=2, 9000, for(y=1, 9000, if(sigma(x+y-1)>(x*y)&&sigma(sigma(x+y-1)-(x*y))==x,print(x-1," ",y)))) $\endgroup$
    – user142929
    Jun 27, 2020 at 23:32
  • $\begingroup$ One can to state similar (or variants) conjectures involving other arithmetic functions. For example denoting the Dedekind psi function as $\psi(n)$ and inspired in my question posted on Mathematics Stack Exchange with identificator 3728632, I wrote that if $x\geq 2$ and $y\geq 1$ are integers that satisfy $$\psi(2(\psi(xy)-(x+y-1))-1)=xy$$ then $x-1$ is a Mersenne prime and $y=1$. $\endgroup$
    – user142929
    Jun 29, 2020 at 20:59
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    $\begingroup$ The referenced MSE question: math.stackexchange.com/questions/3728632/… . $\endgroup$
    – LSpice
    Jul 28, 2020 at 1:28
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    $\begingroup$ Many thanks @LSpice for pointing the references for MSE or in other posts of mine (in past ocassions) for the PSE site. I didn't add this message previously but I always appreciate it. One of my more recent and unanswered question in Physics Stack Exchange has title Wind turbines or tidal power stations incorporating elements of origami or reconfigurable materials: is innovation possible from physics?, with identificator 574543 I add it as invitation if you want to read it. $\endgroup$
    – user142929
    Sep 7, 2020 at 22:29
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    $\begingroup$ Re, No worries. The referenced PSE question: physics.stackexchange.com/questions/574543/… . Note that, if you go to the trouble of remembering the ID, then there's no need to remember the slug: physics.stackexchange.com/questions/574543 works just as well (in fact, physics.stackexchange.com/questions/574543/this-slug-is-ignored also works). $\endgroup$
    – LSpice
    Sep 7, 2020 at 22:42

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